Texture Node in the cycle render
The texture (usually called a diffuse map) simply affects the color of the object.  The normal map affects the normals to create a bump-mapping effect.  This works by converting the red and green channels of the normal map to x and y (or y and x, I forget which) rotation of the surface normals (0.5 being straight up, or no change) to simulate small bumps and surface details for added realism.  (The blue channel is actually ignored, though it is usually set at 0.5 as well which makes for the distinctive pinkish/bluish color of a normal map.)

As for your setup, you have two problems:


You need to send the normal map through the color input of a vector > normal map node and the normal output of that to the normal input of the shaders.
You are giving the normal map normals as mapping coordinates, all texture maps should have the same mapping coordinates.  In this case it looks like you are using generated for the diffuse so you should use generated coordinates for the normal map as well.



Credit to @Vladimir for image, this is a croped version of his screenshot as I don't have Blender available at the moment.